A suitor is a man who seeks to court a woman in the hopes of achieving a serious relationship and possibly marriage. In Miss Emily's case, while her father was alive, he discouraged her prospective suitors because, according to many of the townspeople,

... the Griersons held themselves a little too high for what they really were. None of the young men were quite good enough for Miss Emily and such.

Simply put, Mr. Grierson believed that Emily deserved better than the men of Jefferson, and he must not have allowed any of her relationships with young men to become too serious. By the time Miss Emily had become 30 and was still unmarried (a very late age for a woman to still be without a husband in the late 19th century), some people in Jefferson wondered if any serious marital prospects had ever really materialized. By the time she finally met Homer Barron, Miss Emily was practically an "old maid"--a spinster--and she may have seen Homer as her last chance to ever find true love and marriage.